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Mathematics - Probability Theory and Stochastic Processes | Brownian Motion and its Applications to Mathematical Analysis - École d'Été de Probabilités de

Brownian Motion and its Applications to Mathematical Analysis

École d'Été de Probabilités de Saint-Flour XLIII – 2013

Burdzy, Krzysztof

2014, XII, 137 p. 16 illus., 4 illus. in color.

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  • Contains interesting examples of couplings
  • Gentle introduction to Brownian motion and analysis
  • Heuristic explanations of the main results
These lecture notes provide an introduction to the applications of Brownian motion to analysis and, more generally, connections between Brownian motion and analysis. Brownian motion is a well-suited model for a wide range of real random phenomena, from chaotic oscillations of microscopic objects, such as flower pollen in water, to stock market fluctuations. It is also a purely abstract mathematical tool which can be used to prove theorems in "deterministic" fields of mathematics.
The notes include a brief review of Brownian motion and a section on probabilistic proofs of classical theorems in analysis. The bulk of the notes are devoted to recent (post-1990) applications of stochastic analysis to Neumann eigenfunctions, Neumann heat kernel and the heat equation in time-dependent domains.

Content Level » Research

Keywords » "hot spots" conjecture - 60J65, 60H30, 60G17 - Brownian motion - Neumann eigenfunction - coupling - heat equation

Related subjects » Analysis - Dynamical Systems & Differential Equations - Probability Theory and Stochastic Processes

Table of contents 

1. Brownian motion.- 2. Probabilistic proofs of classical theorems.- 3. Overview of the "hot spots" problem.- 4. Neumann eigenfunctions and eigenvalues.- 5. Synchronous and mirror couplings.- 6. Parabolic boundary Harnack principle.- 7. Scaling coupling.- 8. Nodal lines.- 9. Neumann heat kernel monotonicity.- 10. Reflected Brownian motion in time dependent domains.

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